[Proof]
Motivated by the works of [1]} and [2]}, multiplying \( (\ref {raodong})_1 \)  by \( -R \bar{\theta } \left( \frac{1}{v} - \frac{1}{\bar{v}} \right) \) , \( (\ref {raodong})_2 \)  by \( \psi \)  and \( (\ref {raodong})_3 \)  by \( \zeta \theta ^{-1} \)  respectively, and adding up the resulting equations, after some detailed calculation, we obtain
\( &\quad \left(\frac{1}{2} \psi ^{2} + R \bar{\theta } \Phi \left(\frac{v}{\bar{v}}\right) + \frac{R}{\gamma -1} \bar{\theta } \Phi \left(\frac{\theta }{\bar{\theta }}\right)\right)_{t} +\mu \frac{\psi _{x}^{2}}{v} +\kappa \frac{\zeta _{x}^{2}}{v \theta } + H_{x} \nonumber \\[2mm]&= F \psi + Q + v(E+\psi b+\bar{u}b)^2\frac{\zeta }{\theta } -v(E+\psi b+\bar{u}b)\psi b, \) 
